Time in physics and biology

In contrast with classical physics, particularly with Sir Isaac Newton, where time is a continuous function, generally valid, eternally and evenly flowing as an absolute time dimension, in the biological sciences, time is in essence of cyclical nature (physiological periodicities), where future passes to past through an infinitely thin boundary, the present. In addition, the duration of the present (DP) leads to the so-called 'granulation of time' in living beings, so that by the fusion of two successive pictures of the world, which are not entirely similar, they attain the perception of 'movement,' both in the real world as well as in the sham-movement in the mass media (TV). Key terms: allometric equations, biological time, body mass, physical time, theory of biological similitude.


INTRODUCTION
The apparent antinomy concerning 'time's-arrow' and 'time's-cycle' has already been discussed thoroughly by S. J. Gould (1987), particularly the meaning of time in the geological sciences.It should be recalled that the paradigm of 'time's-arrow' is prevalent in physics, as exemplified in Price's (1996) monograph on the subject, while the concept of 'time's cycle' is commonly applied as a metaphor in the biological sciences.

TIME IN PHYSICS
Since Newton's research on time, more specifically in his second law of motion, the quantitative analysis of the time dimension (T) was incorporated into his opera magna, "On Rational Mechanics," along with the dimensions of mass (M) and length (L).It should be emphasized that the metaphor of 'time's arrow' is already present in Newton's stereotypic definition: "Tempus absolutum, verum et mathematicum." In 1915 Albert Einstein introduced his theory of general relativity into modern physics, and therefore the dimension of time lost its Newtonian characteristic invariance.'Time's-arrow' of the physical sciences has the dimension of a period (T), whose 'duration' can be determined at the present time with an outermost precision (Gibbs, 2002), from the duration of one period of an atomic clock, which is 10 -17 seconds to the astronomical time duration of billions of years, the other extreme of the time spectrum of the physical sciences.

TIME IN BIOLOGY
In the biological sciences the time dimension has a cyclical character, which is commonly expressed as a frequency or the number of complete cycles per unit time (Hertz, Hz).There are many of these relevant chronobiological variables, including, for instance: heart rate, respiratory rate, metabolic rate, gastrointestinal periodicities, renal clearances, muscular activity, electroencephalographic rhythms and nictimeral cycles.For this very reason, the `duration of a single period' of any of these biological rhythms may be of paramount scientific interest, but it is a misleading approach from a strictly biological point of view.In consequence, Calder's (1984) 'physiological time' or Schmidt-Nielsen's (1997) 'metabolic time,' which commonly appears in the biological literature, are two expressions that pertain to the physical realm and not to biology.

GRAVITY AS 'ZEITGEBER' ('Zeitgeber' is a term of German origin and means "time indicator" or "indicator of time.")
In a recent review entitled "A matter of time," Gibbs (2002) discussed the relationship between gravity and time and concluded that the stronger the gravitational pull, the slower time passes, as can be exemplified by the fact that a clock at the top of Mount Everest pulls ahead of those clocks at sea level by about 30 microseconds a year.The same author has added that raising a clock by 10 centimeters will change its rate by one part in 10 17 .
In the case of all living beings, their body weights (W) are the consequence of the above mentioned force exerted on the respective body masses by the constant gravitational field of the planet Earth.On the contrary, when this gravitational force is absent or almost absent (microgravity), the physiological consequences are dramatic, as has been exemplified in space medicine.
In 1687 Newton formulated his second law of motion (F = m•a).In the biological sciences, body mass (M) should have been differentiated from body weight (W), due to the fact that the former is a physical entity, which is first related to the quantity of matter that holds together in one body, and second to the concept of inertial resistance.Furthermore, 'body weight' is primarily of empirical nature and commonly associated with the procedure of weighting organisms by means of a balance and represents a measurement of the force (F) with which a body is attracted toward the center of the earth.
By means of the dimensional analysis and the subsequent development of several theories of biological similarity it has been possible to elucidate -even in a quantitative manner -the physical dilemma mentioned above (mass vs. weight).In consequence, from the classical dimensional analysis (MLT-system of physics) many biological variables can be defined as the product of three power functions: When Lambert and Teissier (1927) established the first theory of biological similarity, they postulated that first, the prototype (p) and the model (m) should have the same density, and second, that the times ratio should vary in proportion to the lengths ratio, both in prototype (p) and model (m), yielding:

MASS AND WEIGHT
The difference between the physical dimensions of mass (M) and weight (W) can be centered on the absence or the presence of the acceleration of gravity (g), because (W = M•g), and for this reason, the dimensional analysis (DA) provides a correct answer, since ] which is finally equivalent to the dimension of a 'force' in the physical sciences.As was already suggested by Newton, instead of the three power functions mentioned above (Eqn.1), it is possible to express the great majority of biological variables only as function of body mass (M) or of body weight (W), with the purpose to be able to compare the 'theoretically' calculated reduced exponent (b) with the `empirical findings,' which are commonly expressed in the form of Huxley's (1932) allometric equation: or after a log-log transformation: The mechanical similarity of classical physics (Galileo and Newton) yielded an allometric exponent of 1/6 for the each oscillation of a pendulum, whereas Lambert and Teissier (1927) proposed an a priori exponent of 1/3 for the time exponent of biological functions.Sernetz et al. (1985), West et al. (1997) and Günther and Morgado (2003a) later proposed the 1/4 exponent for biological rhythms, considering the fractal nature of biological time (See Table I and Fig. 1).
By means of dimensional analysis (Günther and Morgado, 2003b) we have proposed the following two solutions for the allometric exponents (b): 1) when expressed as function of body mass (M) 2) or, as a function of body weight (W) The difference between these calculated allometric exponents (b) is due to the fact that [a] = [L•T -2 ], the equivalent mass exponent for the length dimension (L) is 1/3•β, and for the time dimension is -2/4•γ, which finally yields 1/3 -2/4 = -1/6 = -0.17.
It is worth mentioning that there are two characteristics of holistic nature of paramount importance in each living being; one is body mass (M) or body weight (W), and the other is metabolic rate (MR), which can be measured either as oxygen consumption (liters/min), heat production (kcal/hr), or by measuring the ingestion of some metabolites (carbohydrates and fat).As a paradigm of the metabolic rate per day of an adult and resting human being (70 kg of body weight), one can mention the following variables: 1) Oxygen consumption ≈ 350 liters; 2) Heat production ≈ 1600 kcal; 3) Synthesis and consumption of ATP in all mitochondria of the body ≈ 100 kg; 4) Glucose catabolism ≈ 400 gram; and 5) Power output ≈ 80 watt.
With the purpose of calculating the theoretical allometric exponent (b) for the metabolic rate (MR) we should know first the corresponding physical dimensions: which yielded, according to equations 5 and 6, the following values: 1) as a function of body mass (M) we have: 2) as a function of body weight (W) we obtain: Thus, the difference between both allometric exponents (b) is ∠b = 11/12 _ 3/4 = 1/6 = 0.17, the same figure (0.17) we have already mentioned with regard to the influence of the acceleration of gravity (g) with regard to body mass (M).

TABLE I
The duration of several physiological periods (Calder, 1984) and the corresponding parameters for the frequency spectrum as shown in Figure 1.Página 4 de 8 The empirical counterpart of the above mentioned theoretical assumptions (weight vs. mass) is based on the heart rate (HR) changes during the fetal life, in this case due to the effects of buoyancy, as compared to those of the postnatal period.The corresponding allometric equations are: The difference is -0.21 _ (-0.04) = -0.17,which is the same value we found in the former theoretical approach.In the present comparison, the other relevant variable is the metabolic rate (V O2 ), which is given in ml/min).Mortola (2001) has who found that V O2 in adult mammals is a function of ~W0.75 , while during fetal life V O2 ~ W 0.92 , an empirical finding which agrees with the above mentioned theoretical assumptions (Eqs.7 and 8).

DURATION OF THE PRESENT AND FLICKER FUSION FREQUENCY
The above mentioned fractal nature of biological time (T α M 1/4 α W 1/4 ) raises the question concerning the duration of each of these biological cycles.The duration of the present (DP) has been extensively discussed by Efron (1967) and refers to the period of time during which we are aware of an event or an entity.The existence of a temporal processing period can be exemplified by two different types of sensory perceptions in humans, as for instance by means of tachistoscopic experiments.When the experimental subject is exposed to a flash of red light of only 20 milliseconds (ms) duration, immediately followed by a 20 ms green flash, he reported seeing a flash of yellow light.The interesting fact is that the observer is not aware of the first red light flash, nor of the following 20 ms green light flash; however, if both are given together in immediate succession, the integrative process in the central nervous system (SNC) is successful.On the other hand, a 20 ms burst of a 1000 Hz tone, immediately followed by a 20 ms burst of 4000 Hz tone, is experienced as similar in quantity to a 40 ms burst of a mixture of these two tones (2500 Hz).Thus, the above mentioned author came to the conclusion, that the duration of the present (DP) is no longer than 60 _ 70 ms, or else, it is equivalent to 16.7 -14.3 Hz.In essence, all physiological rhythmicity is of a fractal nature, particularly the heart rate (Goldberger, 1997).
The corresponding allometric equations are for: 1) the duration of the present (DP) in humans of different ages where the duration of the present (DP) is given in (ms) and the body weight (W) in kg, and 2) the flicker-fusion-frequency where (fff) is given in (Hz) and (W) in kg.
On the other hand, it is worth mentioning that the relevance of these chronobiological data is directly concerned with cinematography and television (TV), where approximately 30 pictures per second are visible as a continuous event, where 91.5% of time corresponds to a given image exposure and 8.5% to the black interval between two subsequent pictures.A one-yearold child (Table II, item 2) has an estimated fff of 28.1 Hz, and in consequence, the TV screen is suitable for a one-year-old (30 pictures/sec), because at this frequency the flickering picture has been avoided.Página 6 de 8

DISCUSSION
The two complementary views of time have been discussed already by Gould (1987) and in particular with regard to the analysis of 'geological time.'The proposed dichotomy can also be applied to the essence of time in the physical sciences.The same author has suggested the symbol of the 'time arrow' or the one-way forward flow of time as the metaphor for a particular and unrepeatable event, just as history marks each moment of time with a distinct brand.On the other hand, Gould's metaphor for the concept of 'time cycles' is directly applicable in the biological sciences where time has no direction since it is always present and never changing and can furthermore be characterized by the repetition of a perpetual oscillating series of events.
In the biological realm, and according to Schaltenbrand (1967), the past and the future overlaps during the present.

SUMMARY
1.-The dimension of time in the physical sciences has been characterized by means of Gould's metaphor of 'time's-arrow,' with a wide chronometric spectrum that comprises the duration of a cycle of an atomic clock at one end and on the other extreme, the billion years of the entire universe.
2.-Concerning the biological sciences, the metaphor of the metric of time is Gould's 'time'scycle,' whose quantitative expression by means of Huxley's allometric equation (Y = a•X b ) yields always an exponent b = -0.25,due to the fractal nature of biological time.
3.-From dimensional analysis and one of the theories of biological similarity it has been possible to investigate the influence of earth's gravity on many functions (heart rate and metabolic rate, among others), when fetal body mass (M) and postnatal body weight (W) are utilized as reference systems.
4.-The dimension of time in physics can be defined as a continuous and evenly flowing event of general validity, in contrast with the concept of time in biology, where the predominance of the cyclic nature of almost all functions, leads to the 'granulation' of time, which in the neurophysiological realm appears as 'flicker-fusion-frequencies' and in organ physiology as heart rate, respiratory rate, and specific metabolic rate, among many others iterative functions, in agreement with Fraser's (1966) dictum: "Cycle repetition is the rule of living." 5.-The exponent (b) of Huxley's allometric equation can be predicted theoretically for the great majority of biological functions, based on the dimensional analysis of each variable.The theoretical exponent (b) can be compared with the empirical findings, after the corresponding statistical analysis of the experimental data, which also yields the numerical value of parameter (a) of the above mentioned equation.Finally, the log-log plot of the parameter (a) makes it possible to precisely localize the frequency domain of each physiological variable, whose universal fractal slope is b = -0.25.

Figure 1 .
Figure 1.Double logarithmic plot of five frequency domains of empirical nature (Calder, 1984) as compared with the mean theoretical reduced exponent (b).

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Calder III WA (1984) Size, Function, and Life History.Cambridge, MA: Harvard University Press